In Secs.  and  we saw how the bundles \(\mathrm {Spec}(V)\)  and \(\operatorname{Eig}(V)\)  provide a holonomy interpretation for the physics of geometric phases and exceptional points. This concerns holonomy of semi-principal bundles, which is different from the holonomy of principal bundles. In particular, we did not describe the holonomy operations explicitly using holonomy matrices. Our goal in this section is to move to the corresponding frame bundles, and show how holonomy matrices appear naturally on these. For example, the holonomy around an EP2 is specified by a pattern \(\psi _1 \rightarrow \psi _2 \rightarrow -\psi _1\)  [1]}, and we will formally associate this with the matrix
\( M_{(\psi _1,\psi _2)}=\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}.\) 
